5oo Feathering Paddle- Wheel. or z : z1 : : a-^ : a, and the triangles are similar, so that angle a ~ angle /3. But a is a right angle, being in a semicircle. /. Angle ft is always a right angle, and pr is a straight line. Scott-Russell's motion 5, Fig. 447, merely copies at A D the truth of the slide c, D A c being always a right angle. A more convenient form, is the Grasshopper motion, Fig. 467, where the slide is replaced by a long link. The gear may be formed (i) with AB = BC = BD as in Fig. 447, or (2) A B : B c : : B c : B D, the second being used in grasshopper engines and the first in a steam crane built by Messrs. R. & W. Hawthorn, where a piston connects directly with r> to lift the load. The relation of the links in case (i) may be found graphically : produce points D, B, c, to the respective positions i, 2, 3, on the base line i, 3,: with centre 2 strike arcs i, 4, and 3, 5 : join 4, 5, and draw 5, 6, at right angles to 5, 4. Then 6 produced gives point A, and length of A B ; for 5, 2 is a mean proportional between 6, 2 and 2, 4. The Feathering Paddle-Wheel is shewn in Fig. 468. If the vessel move to the right with a velocity vs while the wheel rim has a linear velocity of v/t the floats should enter and leave the water in the directions vr if they are to meet the water with- out shock, for vr is the relative velocity of float to water, found by completing the parallelogram. The controlling mechanism is obtained by quadric chain H G K E where H G is the fixed link. Stresses iri Linkwork Members may be ascertained from the principles in Fig. 423 et seq., the structure being balanced by known external forces. The Work Done at any point of a machine is obtained as at Fig. 325. Taking the case of harmonic motion for donkey pump, let total piston pressure P be uniform during stroke d': then P*/' = work done at P and is shewn by diagram in Fig. 469. Setting out trie pressure-curve for W, on a base ?r R, as explained in Fig. 453, tne mean of the ordinates will be found to be •636 P, and as , . work put in = work taken out P X 2R » '636 P x TrR